Idea

Naïve vs axiomatic set theory

Naïve set theory is the basic algebra of the subsets of any given set UU, together with a few levels of power sets, say up to 𝒫𝒫𝒫U\mathcal{P}\mathcal{P}\mathcal{P}U and possibly no further. Often students see this first for the set of real numbers as UU (although in fact one could start with the set of natural numbers and go one level further for an equivalent theory). One could also use function sets instead of power sets as the basic set-forming operation (especially for a weakly predicative theory in constructive mathematics).

Once you start thinking very much about the nature of sets in general (rather than merely using naïve set theory), it quickly becomes clear that one must be careful about how one can and cannot form sets. Georg Cantor is credited with being the first to think about sets this deeply; although he did not propose a system of general rules for valid set-making operations, he recognised that some sets were ‘inconsistent’. An axiomatic set theory is a set theory which carefully states the rules (or ‘axioms’) which sets are assumed to obey.

Foundational vs definitional set theory

There are two ways to go about doing axiomatic set theory. The more ambitious is to develop a foundational set theory: set theory as foundations for all of mathematics. This is what Frege proposed (although he failed through inconsistency) and which Zermelo achieved.

A variation of Zermelo's system (developed by Fraenkel and Skolem and called Zermelo–Fraenkel set theory or ZFC) is the orthodox foundations today, although it needs to be supplemented by Grothendieck universes (or something along those lines) to handle modern category theory, and set theorists often consider further strengthenings of it through large cardinalaxioms (of which the existence of Grothendieck universes is an example, just the tip of the iceberg).

Material vs structural set theory

On the nnLab we like to distinguish between two types of set theory, especially in foundations:

In a material set theory (also called a membership-based set theory), the elements of a set exist independently of that set. As such, it makes sense to take two completely unrelated sets and ask if two of their members are equal, and sometimes the answer will be Yes. Frequently in material set theory one takes everything to be a pure set, so that elements of sets are themselves sets. Therefore, any two sets may be meaningfully compared to ask if they are equal or if one is a member of the other.

A structural set theory, on the other hand, looks more like type theory. Here, the elements of a set have no existence or structure apart from their identity as elements of that set. In particular, they are not themselves sets, and cannot be elements of any other set, at least not without invoking some explicit type-casting? operator (which here could simply be a function from one set to the other). Similarly, elements of different sets cannot be compared to each other (without type-casting them to become elements of the same set).

Among category theorists, it's popular to state the axioms of a structural set theory by specifying elementary properties of the category of sets; the orthodoxy here (to the extent that there is one) is probably Bill Lawvere's ETCS, which suffices for most everyday uses but must be supplemented to handle some esoteric parts of modern mathematics. Another structural set theory, which is stronger than ETCS and less closely tied to category theory, is SEAR.

In contrast, ZFC is an example of a material set theory. From a model of either kind of set theory we can construct a model of the other, so the two are, broadly speaking, equivalent; for example, SEARC (SEAR with the axiom of choice) is equivalent in this way to ZFC, while ETCS is equivalent to a weak (‘bounded’) variation of ZFC. A more precise statement is that the two kinds of theories form categories related by the material-structural adjunction.

Category theory on set theory

Category theory can provide a common model theory to compare various set theories. Although only structural set theories like ETCS treat the elementary properties of the category SetSet of sets as fundamental, one can ask for any set theory what properties SetSet satisfies and compare them in those terms. At the very least, SetSet should be a pretopos.